Theorem bj-eldiag 37503

Description: Characterization of the elements of the diagonal of a Cartesian square. Subsumed by bj-elid6 37497. (Contributed by BJ, 22-Jun-2019.)

Assertion

Ref	Expression
bj-eldiag	⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (Id‘𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st ‘𝐵) = (2nd ‘𝐵))))

Proof of Theorem bj-eldiag

Step	Hyp	Ref	Expression
1		bj-diagval2 37502	. . 3 ⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ∩ (𝐴 × 𝐴)))
2	1	eleq2d 2823	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (Id‘𝐴) ↔ 𝐵 ∈ ( I ∩ (𝐴 × 𝐴))))
3		elin 3906	. . 3 ⊢ (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (𝐵 ∈ I ∧ 𝐵 ∈ (𝐴 × 𝐴)))
4		ancom 460	. . 3 ⊢ ((𝐵 ∈ I ∧ 𝐵 ∈ (𝐴 × 𝐴)) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ 𝐵 ∈ I ))
5		bj-elid4 37495	. . . 4 ⊢ (𝐵 ∈ (𝐴 × 𝐴) → (𝐵 ∈ I ↔ (1st ‘𝐵) = (2nd ‘𝐵)))
6	5	pm5.32i 574	. . 3 ⊢ ((𝐵 ∈ (𝐴 × 𝐴) ∧ 𝐵 ∈ I ) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st ‘𝐵) = (2nd ‘𝐵)))
7	3, 4, 6	3bitri 297	. 2 ⊢ (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st ‘𝐵) = (2nd ‘𝐵)))
8	2, 7	bitrdi 287	1 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (Id‘𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st ‘𝐵) = (2nd ‘𝐵))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ∩ cin 3889   I cid 5516   × cxp 5620  ‘cfv 6490  1^st c1st 7931  2^nd c2nd 7932  Idcdiag2 37499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-iota 6446  df-fun 6492  df-fv 6498  df-1st 7933  df-2nd 7934  df-bj-diag 37500

This theorem is referenced by: (None)